$\left(1-\cot 200°\right)\left(1-\cot 25°\right)=$   

To evaluate the expression (1 - cot 200°)(1 - cot 25°), we can use trigonometric identities and properties of cotangent.

Step 1: Simplify cot(200°)

Recall that:

cot(180° + θ) = cot θ

Thus:

cot 200° = cot (180° + 20°) = cot 20°

The expression becomes:

(1 - cot 20°)(1 - cot 25°)

Step 2: Use the Cotangent Addition Formula

The cotangent addition formula is:

cot(A + B) = (cot A * cot B - 1) / (cot A + cot B)

Let A = 20° and B = 25°. Then:

cot 45° = (cot 20° * cot 25° - 1) / (cot 20° + cot 25°)

Since cot 45° = 1, we have:

1 = (cot 20° * cot 25° - 1) / (cot 20° + cot 25°)

Step 3: Simplify the Equation

cot 20° + cot 25° = cot 20° * cot 25° - 1

Rearrange terms:

cot 20° * cot 25° - cot 20° - cot 25° - 1 = 0

Adding 2 to both sides:

cot 20° * cot 25° - cot 20° - cot 25° + 1 = 2

Factoring:

(1 - cot 20°)(1 - cot 25°) = 2

Final Answer

The value of the expression is 2.